The Hopfield neural network is a discrete time dynamical system composed
of multiple binary nodes, with a connectivity matrix built from a
predetermined set of patterns. The update, inspired from the spin-glass
model (used to describe magnetic properties of dilute alloys), is based on
a random scanning of every node. The existence of a fixed point dynamics
is guaranteed by a Lyapunov function. The Hopfield network is expected to
have those multiple patterns as attractors (multistable dynamical system).
When the initial conditions are close to one of the ‘learned’ patterns,
the dynamical system is expected to relax on the corresponding attractor.
A possible output of the system is the final attractive state (interpreted
as an associative memory).

Various extensions of the initial model have been proposed, among which a
noiseless and continuous version [Hopfield 1984] having a slightly
different Lyapunov function, but essentially the same dynamical
properties, with more straightforward physiological interpretation. A
continuous Hopfield neural network (with a sigmoid transfer function) can
indeed be interpreted as a network of neural masses with every node
corresponding to the mean field activity of a local brain region, with
many bridges with the Wilson Cowan model [WC_1972].

The values for each state-variable should be set to encompass the expected dynamic range of that state-variable for the current parameters, it is used as a mechanism for bounding random inital conditions when the simulation isn’t started from an explicit history, it is also provides the default range of phase-plane plots.

default: {‘x’: array([-1., 2.]), ‘theta’: array([ 0., 1.])}

tauT (\(\tau_{\theta}\))

The slow time-scale for threshold calculus :math:` heta`, state-variable of the model.

default: [ 5.]

range: low = 0.01 ; high = 100.0

taux (\(\tau_{x}\))

The fast time-scale for potential calculus \(x\), state-variable of the model.

default: [ 1.]

range: low = 0.01 ; high = 100.0

variables_of_interest (Variables watched by Monitors)

The values for each state-variable should be set to encompass the expected dynamic range of that state-variable for the current parameters, it is used as a mechanism for bounding random initial conditions when the simulation isn’t started from an explicit history, it is also provides the default range of phase-plane plots.

The Jansen and Rit is a biologically inspired mathematical framework
originally conceived to simulate the spontaneous electrical activity of
neuronal assemblies, with a particular focus on alpha activity, for instance,
as measured by EEG. Later on, it was discovered that in addition to alpha
activity, this model was also able to simulate evoked potentials.

Reciprocal of the time constant of passive membrane and all other spatially distributed delays in the dendritic network [ms^-1]. Also called average synaptic time constant.

default: [ 0.1]

range: low = 0.05 ; high = 0.15

a_1 (\(\alpha_1\))

Average probability of synaptic contacts in the feedback excitatory loop.

default: [ 1.]

range: low = 0.5 ; high = 1.5

a_2 (\(\alpha_2\))

Average probability of synaptic contacts in the slow feedback excitatory loop.

default: [ 0.8]

range: low = 0.4 ; high = 1.2

a_3 (\(\alpha_3\))

Average probability of synaptic contacts in the feedback inhibitory loop.

default: [ 0.25]

range: low = 0.125 ; high = 0.375

a_4 (\(\alpha_4\))

Average probability of synaptic contacts in the slow feedback inhibitory loop.

default: [ 0.25]

range: low = 0.125 ; high = 0.375

b (\(b\))

Reciprocal of the time constant of passive membrane and all other spatially distributed delays in the dendritic network [ms^-1]. Also called average synaptic time constant.

default: [ 0.05]

range: low = 0.025 ; high = 0.075

mu (\(\mu_{max}\))

Mean input firing rate

default: [ 0.22]

range: low = 0.0 ; high = 0.22

nu_max (\(\nu_{max}\))

Determines the maximum firing rate of the neural population [s^-1].

default: [ 0.0025]

range: low = 0.00125 ; high = 0.00375

p_max (\(p_{max}\))

Maximum input firing rate.

default: [ 0.32]

range: low = 0.0 ; high = 0.32

p_min (\(p_{min}\))

Minimum input firing rate.

default: [ 0.12]

range: low = 0.0 ; high = 0.12

r (\(r\))

Steepness of the sigmoidal transformation [mV^-1].

default: [ 0.56]

range: low = 0.28 ; high = 0.84

state_variable_range (State Variable ranges [lo, hi])

The values for each state-variable should be set to encompass the expected dynamic range of that state-variable for the current parameters, it is used as a mechanism for bounding random inital conditions when the simulation isn’t started from an explicit history, it is also provides the default range of phase-plane plots.

Firing threshold (PSP) for which a 50% firing rate is achieved. In other words, it is the value of the average membrane potential corresponding to the inflection point of the sigmoid [mV]. The usual value for this parameter is 6.0.

default: [ 5.52]

range: low = 3.12 ; high = 6.0

variables_of_interest (Variables watched by Monitors)

This represents the default state-variables of this Model to be monitored. It can be overridden for each Monitor if desired. The corresponding state-variable indices for this model are \(y0 = 0\), \(y1 = 1\), \(y2 = 2\), \(y3 = 3\), \(y4 = 4\), and \(y5 = 5\)

Maximum firing rate to the pyramidal population [ms^-1]. (External stimulus. Constant intensity.Entry point for coupling.)

default: [ 0.12]

range: low = 0.0 ; high = 0.35

Q (\(Q\))

Maximum firing rate to the interneurons population [ms^-1]. (External stimulus. Constant intensity.Entry point for coupling.)

default: [ 0.12]

range: low = 0.0 ; high = 0.35

U (\(U\))

Maximum firing rate to the stellate population [ms^-1]. (External stimulus. Constant intensity.Entry point for coupling.)

default: [ 0.12]

range: low = 0.0 ; high = 0.35

e0 (\(e_0\))

Half of the maximum population mean firing rate [ms^-1].

default: [ 0.0025]

range: low = 0.00125 ; high = 0.00375

gamma_1 (\(\gamma_1\))

Average number of synapses between populations (pyramidal to stellate).

default: [ 135.]

range: low = 65.0 ; high = 1350.0

gamma_1T (\(\gamma_{1T}\))

Coupling factor from the extrinisic input to the spiny stellate population.

default: [ 1.]

range: low = 0.0 ; high = 1000.0

gamma_2 (\(\gamma_2\))

Average number of synapses between populations (stellate to pyramidal).

default: [ 108.]

range: low = 0.0 ; high = 200

gamma_2T (\(\gamma_{2T}\))

Coupling factor from the extrinisic input to the pyramidal population.

default: [ 1.]

range: low = 0.0 ; high = 1000.0

gamma_3 (\(\gamma_3\))

Connectivity constant (pyramidal to interneurons)

default: [ 33.75]

range: low = 0.0 ; high = 200

gamma_3T (\(\gamma_{3T}\))

Coupling factor from the extrinisic input to the inhibitory population.

default: [ 1.]

range: low = 0.0 ; high = 1000.0

gamma_4 (\(\gamma_4\))

Connectivity constant (interneurons to pyramidal)

default: [ 33.75]

range: low = 0.0 ; high = 200

gamma_5 (\(\gamma_5\))

Connectivity constant (interneurons to interneurons)

default: [15]

range: low = 0.0 ; high = 100

ke (\(\kappa_e\))

Reciprocal of the time constant of passive membrane and all other spatially distributed delays in the dendritic network [ms^-1]. Also called average synaptic time constant.

default: [ 0.1]

range: low = 0.05 ; high = 0.15

ki (\(\kappa_i\))

Reciprocal of the time constant of passive membrane and all other spatially distributed delays in the dendritic network [ms^-1]. Also called average synaptic time constant.

default: [ 0.05]

range: low = 0.025 ; high = 0.075

rho_1 (\(\rho_1\))

Steepness of the sigmoidal transformation [mV^-1].

default: [ 0.56]

range: low = 0.28 ; high = 0.84

rho_2 (\(\rho_2\))

Firing threshold (PSP) for which a 50% firing rate is achieved. In other words, it is the value of the average membrane potential corresponding to the inflection point of the sigmoid [mV]. Population mean firing threshold.

default: [ 6.]

range: low = 3.12 ; high = 10.0

state_variable_range (State Variable ranges [lo, hi])

The values for each state-variable should be set to encompass the expected dynamic range of that state-variable for the current parameters, it is used as a mechanism for bounding random inital conditions when the simulation isn’t started from an explicit history, it is also provides the default range of phase-plane plots.

This represents the default state-variables of this Model to be monitored. It can be overridden for each Monitor if desired. The corresponding state-variable indices for this model are \(v_6 = 0\), \(v_7 = 1\), \(v_2 = 2\), \(v_3 = 3\), \(v_4 = 4\), and \(v_5 = 5\)

Equations and default parameters are taken from [Breaksetal_2003_b].
All equations and parameters are non-dimensional and normalized.
For values of d_v < 0.55, the dynamics of a single column settles onto a
solitary fixed point attractor.

Parameters used for simulations in [Breaksetal_2003_a] Table 1. Page 153.
Two nodes were coupled. C=0.1

Strength of excitatory coupling. Balance between internal and local (and global) coupling strength. C > 0 introduces interdependences between consecutive columns/nodes. C=1 corresponds to maximum coupling between node and no self-coupling. This strenght should be set to sensible values when a whole network is connected.

Variance of the excitatory threshold. It is one of the main parameters explored in [Breaksetal_2003_b].

default: [ 0.65]

range: low = 0.49 ; high = 0.7

d_Z (\(\delta_{Z}\))

Variance of the inhibitory threshold.

default: [ 0.7]

range: low = 0.001 ; high = 0.75

gCa (\(g_{Ca}\))

Conductance of population of Ca++ channels.

default: [ 1.1]

range: low = 0.9 ; high = 1.5

gK (\(g_{K}\))

Conductance of population of K channels.

default: [ 2.]

range: low = 1.95 ; high = 2.05

gL (\(g_{L}\))

Conductance of population of leak channels.

default: [ 0.5]

range: low = 0.45 ; high = 0.55

gNa (\(g_{Na}\))

Conductance of population of Na channels.

default: [ 6.7]

range: low = 0.0 ; high = 10.0

phi (\(\phi\))

Temperature scaling factor.

default: [ 0.7]

range: low = 0.3 ; high = 0.9

rNMDA (\(r_{NMDA}\))

Ratio of NMDA to AMPA receptors.

default: [ 0.25]

range: low = 0.2 ; high = 0.3

state_variable_range (State Variable ranges [lo, hi])

The values for each state-variable should be set to encompass the expected dynamic range of that state-variable for the current parameters, it is used as a mechanism for bounding random inital conditions when the simulation isn’t started from an explicit history, it is also provides the default range of phase-plane plots.

The Generic2dOscillator model is a generic dynamic system with two state
variables. The dynamic equations of this model are composed of two ordinary
differential equations comprising two nullclines. The first nullcline is a
cubic function as it is found in most neuron and population models; the
second nullcline is arbitrarily configurable as a polynomial function up to
second order. The manipulation of the latter nullcline’s parameters allows
to generate a wide range of different behaviours.

Constant parameter to scale the rate of feedback from the slow variable to the fast variable.

default: [ 1.]

range: low = -5.0 ; high = 5.0

b (\(b\))

Linear slope of the configurable nullcline

default: [-10.]

range: low = -20.0 ; high = 15.0

beta (\(\beta\))

Constant parameter to scale the rate of feedback from the slow variable to itself

default: [ 1.]

range: low = -5.0 ; high = 5.0

c (\(c\))

Parabolic term of the configurable nullcline

default: [ 0.]

range: low = -10.0 ; high = 10.0

d (\(d\))

Temporal scale factor. Warning: do not use it unless you know what you are doing and know about time tides.

default: [ 0.02]

range: low = 0.0001 ; high = 1.0

e (\(e\))

Coefficient of the quadratic term of the cubic nullcline.

default: [ 3.]

range: low = -5.0 ; high = 5.0

f (\(f\))

Coefficient of the cubic term of the cubic nullcline.

default: [ 1.]

range: low = -5.0 ; high = 5.0

g (\(g\))

Coefficient of the linear term of the cubic nullcline.

default: [ 0.]

range: low = -5.0 ; high = 5.0

gamma (\(\gamma\))

Constant parameter to reproduce FHN dynamics where excitatory input currents are negative. It scales both I and the long range coupling term.

default: [ 1.]

range: low = -1.0 ; high = 1.0

state_variable_range (State Variable ranges [lo, hi])

The values for each state-variable should be set to encompass the expected dynamic range of that state-variable for the current parameters, it is used as a mechanism for bounding random initial conditions when the simulation isn’t started from an explicit history, it is also provides the default range of phase-plane plots.

default: {‘W’: array([-6., 6.]), ‘V’: array([-2., 4.])}

tau (\(\tau\))

A time-scale hierarchy can be introduced for the state variables \(V\) and \(W\). Default parameter is 1, which means no time-scale hierarchy.

default: [ 1.]

range: low = 1.0 ; high = 5.0

variables_of_interest (Variables or quantities available to Monitors)

The quantities of interest for monitoring for the generic 2D oscillator.

The Kuramoto model is a model of synchronization phenomena derived by
Yoshiki Kuramoto in 1975 which has since been applied to diverse domains
including the study of neuronal oscillations and synchronization.

\(\omega\) sets the base line frequency for the Kuramoto oscillator in [rad/ms]

default: [ 1.]

range: low = 0.01 ; high = 200.0

state_variable_range (State Variable ranges [lo, hi])

The values for each state-variable should be set to encompass the expected dynamic range of that state-variable for the current parameters, it is used as a mechanism for bounding random initial conditions when the simulation isn’t started from an explicit history, it is also provides the default range of phase-plane plots.

default: {‘theta’: array([ 0. , 6.28318531])}

variables_of_interest (Variables watched by Monitors)

This represents the default state-variables of this Model to be monitored. It can be overridden for each Monitor if desired. The Kuramoto model, however, only has one state variable with and index of 0, so it is not necessary to change the default here.

Calculate coefficients for the Reduced FitzHugh-Nagumo oscillator based
neural field model. Specifically, this method implements equations for
calculating coefficients found in the supplemental material of
[SJ_2008].

Include equations here...

#NOTE: In the Article this modelis called StefanescuJirsa2D

traits on this class:

K11 (\(K_{11}\))

Internal coupling, excitatory to excitatory

default: [ 0.5]

range: low = 0.0 ; high = 1.0

K12 (\(K_{12}\))

Internal coupling, inhibitory to excitatory

default: [ 0.15]

range: low = 0.0 ; high = 1.0

K21 (\(K_{21}\))

Internal coupling, excitatory to inhibitory

default: [ 0.15]

range: low = 0.0 ; high = 1.0

a (\(a\))

doc...

default: [ 0.45]

range: low = 0.0 ; high = 1.0

b (\(b\))

doc...

default: [ 0.9]

range: low = 0.0 ; high = 1.0

mu (\(\mu\))

Mean of Gaussian distribution

default: [ 0.]

range: low = 0.0 ; high = 1.0

sigma (\(\sigma\))

Standard deviation of Gaussian distribution

default: [ 0.35]

range: low = 0.0 ; high = 1.0

state_variable_range (State Variable ranges [lo, hi])

The values for each state-variable should be set to encompass the expected dynamic range of that state-variable for the current parameters, it is used as a mechanism for bounding random inital conditions when the simulation isn’t started from an explicit history, it is also provides the default range of phase-plane plots.

This represents the default state-variables of this Model to be monitored. It can be overridden for each Monitor if desired. The corresponding state-variable indices for this model are \(\xi = 0\), \(\eta = 1\), \(\alpha = 2\), and \(\beta= 3\).

Calculate coefficients for the neural field model based on a Reduced set
of Hindmarsh-Rose oscillators. Specifically, this method implements
equations for calculating coefficients found in the supplemental
material of [SJ_2008].

Include equations here...

#NOTE: In the Article this modelis called StefanescuJirsa3D

traits on this class:

K11 (\(K_{11}\))

Internal coupling, excitatory to excitatory

default: [ 0.5]

range: low = 0.0 ; high = 1.0

K12 (\(K_{12}\))

Internal coupling, inhibitory to excitatory

default: [ 0.1]

range: low = 0.0 ; high = 1.0

K21 (\(K_{21}\))

Internal coupling, excitatory to inhibitory

default: [ 0.15]

range: low = 0.0 ; high = 1.0

a (\(a\))

Dimensionless parameter as in the Hindmarsh-Rose model

default: [ 1.]

range: low = 0.0 ; high = 1.0

b (\(b\))

Dimensionless parameter as in the Hindmarsh-Rose model

default: [ 3.]

range: low = 0.0 ; high = 3.0

c (\(c\))

Dimensionless parameter as in the Hindmarsh-Rose model

default: [ 1.]

range: low = 0.0 ; high = 1.0

d (\(d\))

Dimensionless parameter as in the Hindmarsh-Rose model

default: [ 5.]

range: low = 2.5 ; high = 7.5

mu (\(\mu\))

Mean of Gaussian distribution

default: [ 3.3]

range: low = 1.1 ; high = 3.3

r (\(r\))

Adaptation parameter

default: [ 0.006]

range: low = 0.0 ; high = 0.1

s (\(s\))

Adaptation paramters, governs feedback

default: [ 4.]

range: low = 2.0 ; high = 6.0

sigma (\(\sigma\))

Standard deviation of Gaussian distribution

default: [ 0.3]

range: low = 0.0 ; high = 1.0

state_variable_range (State Variable ranges [lo, hi])

The values for each state-variable should be set to encompass the expected dynamic range of that state-variable for the current parameters, it is used as a mechanism for bounding random inital conditions when the simulation isn’t started from an explicit history, it is also provides the default range of phase-plane plots.

This represents the default state-variables of this Model to be monitored. It can be overridden for each Monitor if desired. The corresponding state-variable indices for this model are \(\xi = 0\), \(\eta = 1\), \(\tau = 2\), \(\alpha = 3\), \(\beta = 4\), and \(\gamma = 5\)

Daffertshofer, A. and van Wijk, B. On the influence of
amplitude on the connectivity between phases
Frontiers in Neuroinformatics, July, 2011

Used Eqns 11 and 12 from [WC_1972] in dfun. P and Q represent external
inputs, which when exploring the phase portrait of the local model are set
to constant values. However in the case of a full network, P and Q are the
entry point to our long range and local couplings, that is, the activity
from all other nodes is the external input to the local population.

External stimulus to the excitatory population. Constant intensity.Entry point for coupling.

default: [ 0.]

range: low = 0.0 ; high = 20.0

Q (\(Q\))

External stimulus to the inhibitory population. Constant intensity.Entry point for coupling.

default: [ 0.]

range: low = 0.0 ; high = 20.0

a_e (\(a_e\))

The slope parameter for the excitatory response function

default: [ 1.2]

range: low = 0.0 ; high = 1.4

a_i (\(a_i\))

The slope parameter for the inhibitory response function

default: [ 1.]

range: low = 0.0 ; high = 2.0

alpha_e (\(\alpha_e\))

External stimulus to the excitatory population. Constant intensity.Entry point for coupling.

default: [ 1.]

range: low = 0.0 ; high = 20.0

alpha_i (\(\alpha_i\))

External stimulus to the inhibitory population. Constant intensity.Entry point for coupling.

default: [ 1.]

range: low = 0.0 ; high = 20.0

b_e (\(b_e\))

Position of the maximum slope of the excitatory sigmoid function

default: [ 2.8]

range: low = 1.4 ; high = 6.0

b_i (\(b_i\))

Position of the maximum slope of a sigmoid function [in threshold units]

default: [ 4.]

range: low = 2.0 ; high = 6.0

c_e (\(c_e\))

The amplitude parameter for the excitatory response function

default: [ 1.]

range: low = 1.0 ; high = 20.0

c_ee (\(c_{ee}\))

Excitatory to excitatory coupling coefficient

default: [ 12.]

range: low = 11.0 ; high = 16.0

c_ei (\(c_{ie}\))

Excitatory to inhibitory coupling coefficient.

default: [ 13.]

range: low = 2.0 ; high = 22.0

c_i (\(c_i\))

The amplitude parameter for the inhibitory response function

default: [ 1.]

range: low = 1.0 ; high = 20.0

c_ie (\(c_{ei}\))

Inhibitory to excitatory coupling coefficient

default: [ 4.]

range: low = 2.0 ; high = 15.0

c_ii (\(c_{ii}\))

Inhibitory to inhibitory coupling coefficient.

default: [ 11.]

range: low = 2.0 ; high = 15.0

k_e (\(k_e\))

Maximum value of the excitatory response function

default: [ 1.]

range: low = 0.5 ; high = 2.0

k_i (\(k_i\))

Maximum value of the inhibitory response function

default: [ 1.]

range: low = 0.0 ; high = 2.0

r_e (\(r_e\))

Excitatory refractory period

default: [ 1.]

range: low = 0.5 ; high = 2.0

r_i (\(r_i\))

Inhibitory refractory period

default: [ 1.]

range: low = 0.5 ; high = 2.0

state_variable_range (State Variable ranges [lo, hi])

The values for each state-variable should be set to encompass the expected dynamic range of that state-variable for the current parameters, it is used as a mechanism for bounding random inital conditions when the simulation isn’t started from an explicit history, it is also provides the default range of phase-plane plots.

default: {‘I’: array([ 0., 1.]), ‘E’: array([ 0., 1.])}

tau_e (\(\tau_e\))

Excitatory population, membrane time-constant [ms]

default: [ 10.]

range: low = 0.0 ; high = 150.0

tau_i (\(\tau_i\))

Inhibitory population, membrane time-constant [ms]

default: [ 10.]

range: low = 0.0 ; high = 150.0

theta_e (\(\theta_e\))

Excitatory threshold

default: [ 0.]

range: low = 0.0 ; high = 60.0

theta_i (\(\theta_i\))

Inhibitory threshold

default: [ 0.]

range: low = 0.0 ; high = 60.0

variables_of_interest (Variables watched by Monitors)

This represents the default state-variables of this Model to be monitored. It can be overridden for each Monitor if desired. The corresponding state-variable indices for this model are \(E = 0\) and \(I = 1\).